When you start learning about trigonometry, you'll meet three important functions: sine, cosine, and tangent.

These functions help us understand how the angles and sides of right triangles relate to each other. Each function has its own definition and purpose. Let's break them down to see how they are different.

Definitions

1. Sine (sin):

   • Sine looks at the opposite side of a triangle compared to the longest side, called the hypotenuse.
   • For an angle ( \theta ), we write this like this: $\text{sin} \, \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
   • For example, imagine a triangle with a 30-degree angle. If the opposite side is 1 unit and the hypotenuse is 2 units, then: $\text{sin} \, 30^\circ = \frac{1}{2}$

2. Cosine (cos):

   • Cosine looks at the side next to the angle (the adjacent side) compared to the hypotenuse.
   • We can express this as: $\text{cos} \, \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
   • Using the same triangle with a 30-degree angle, if the adjacent side is ( \sqrt{3} ) units and the hypotenuse is 2 units, then: $\text{cos} \, 30^\circ = \frac{\sqrt{3}}{2}$

3. Tangent (tan):

   • Tangent connects the opposite side directly to the adjacent side.
   • We write it like this: $\text{tan} \, \theta = \frac{\text{opposite}}{\text{adjacent}}$
   • In our example, if the opposite side is 1 unit and the adjacent side is ( \sqrt{3} ) units, then: $\text{tan} \, 30^\circ = \frac{1}{\sqrt{3}}$
   • We can also write that as: $\frac{\sqrt{3}}{3}$ after simplifying.

Key Differences

• What They Measure:

  • Sine measures the relationship between the opposite side and the hypotenuse.
  • Cosine measures the relationship between the adjacent side and the hypotenuse.
  • Tangent measures the relationship between the opposite side and the adjacent side.

• Visualizing on the Unit Circle:

  • One interesting thing is how these functions relate to the unit circle.
  • On the unit circle, the position of a point related to an angle ( \theta ) is given by: ( (\text{cos} , \theta, \text{sin} , \theta) ).
  • You can think of tangent as the ( y )-coordinate (sine) divided by the ( x )-coordinate (cosine): $\text{tan} \, \theta = \frac{\text{sin} \, \theta}{\text{cos} \, \theta}$

• Usage in Problems:

  • When solving problems about right triangles, which function you use depends on the information you have.
  • If you know two side lengths, you might use sine or cosine.
  • If you have one angle and one side, you might use tangent.

Final Thoughts

Sine, cosine, and tangent are like best friends in trigonometry.

They are essential for solving problems that involve angles and triangles.

With practice, these ideas will start to make sense, and you'll see how they can be used in real life, like in physics and engineering!